Review Session for Final Exam Welcome Scan the QR code below if you want to have open a pdf with the notes I ll be using today Section 4 3 Bases and Dimensions of Subspaces Basis Dimension Ex Let cid 3422 cid 3430 cid 3434 are in cid 3426 Find a basis for 2 3 5 Then determine the dimension of Section 5 1 Eigenvectors and Eigenvalues Definition of an eigenvalue vector Ex Determine if the given vector is an eigenvector of If so find the corresponding eigenvalue Math 220 Fall 2023 Lecture Notes Page 1 1 cid 3429 cid 3433 cid 3429 cid 3433 0 0 2 1 1 2 1 0 3 0 1 0 2 cid 3429 cid 3433 cid 3429 0 0 2 1 2 1 3 1 0 1 2 1 cid 3433 Finding eigenvectors Ex 2 is an eigenvalue of cid 3429 cid 3433 Find a basis for the corresponding eigenspace 0 0 2 1 1 2 1 0 3 Math 220 Fall 2023 Lecture Notes Page 2 Ex 1 is an eigenvalue of cid 3429 cid 3433 Find a basis for the corresponding eigenspace 0 0 2 1 1 2 1 0 3 Eigenvalues of a triangular matrix Section 5 2 The Characteristic Equation Finding eigenvalues Ex Let cid 3429 cid 3433 Find the characteristic equation of 0 0 2 1 2 1 3 1 0 and all eigenvalues Math 220 Fall 2023 Lecture Notes Page 3 Properties of eigenvalues vectors Ex Let cid 3428 4 6 5 3 cid 3432 Find the eigenvalues of 1 cid 2879 cid 2869 2 cid 2872 3 cid 3021 Section 5 3 Diagonalization The diagonalization is an matrix cid 2879 cid 2869 is an diagonal matrix with the eigenvalues of along the diagonal is an matrix whose columns are the corresponding eigenvectors Ex Let 1 Diagonalize 0 0 2 1 2 Math 220 Fall 2023 Lecture Notes Page 4 Ex Let cid 3429 cid 3433 Diagonalize 0 0 2 1 1 2 1 0 3 In earlier examples we found cid 2869 cid 2870 2 cid 2871 1 cid 2869 cid 4652 cid 4652 cid 4652 cid 4652 cid 3429 cid 3433 0 1 0 cid 2870 cid 4652 cid 4652 cid 4652 cid 4652 cid 3429 cid 3433 and cid 2871 cid 4652 cid 4652 cid 4652 cid 4652 cid 3429 cid 3433 1 0 1 2 1 1 Powers of a matrix using diagonalization Ex cid 3429 cid 3433 Find cid 2870 cid 2869 0 0 2 1 1 2 1 0 3 Ex A matrix has eigenvalues cid 2869 3 and cid 2870 1 with corresponding eigenvectors cid 2869 cid 4652 cid 4652 cid 4652 cid 4652 cid 4674 cid 4675 and cid 2870 cid 4652 cid 4652 cid 4652 cid 4652 cid 4674 cid 4675 Find 2 1 1 1 Math 220 Fall 2023 Lecture Notes Page 5 Ex A matrix has eigenvalues cid 2869 3 and cid 2870 1 with corresponding eigenvectors cid 2869 cid 4652 cid 4652 cid 4652 cid 4652 cid 4674 cid 4675 and cid 2870 cid 4652 cid 4652 cid 4652 cid 4652 cid 4674 cid 4675 Find 2 1 1 1 When is a matrix diagonalizable If has distinct eigenvalues If has a repeated eigenvalue with multiplicity Ex Determine if each matrix below is diagonalizable 1 cid 3428 1 3 0 2 cid 3432 2 cid 3428 2 3 0 2 cid 3432 3 cid 3428 2 0 0 2 cid 3432 If cid 2869 1 If cid 2869 1 If cid 2869 1 Long term behavior of population If a population transitions over time according to an matrix then the long term growth rate of the population is Math 220 Fall 2023 Lecture Notes Page 6 The long term population distribution is given by Ex An insect population of Juveniles and Adults is modeled by the matrix cid 3428 5 0 2 30 0 cid 3432 So that each year juveniles reproduce an average of 5 times adults reproduce an average of 30 times 20 of juveniles survive and become adults and no adults survive 1 Suppose that the initial population is 100 juveniles and 100 adults How many juveniles and adults will be in the population after the 1st year 2 Find the long term growth rate of the insect population 2 Find the long term distribution of juveniles and adults as a proportion of the total population Section 6 1 Inner Product Length and Orthogonality Math 220 Fall 2023 Lecture Notes Page 7 Inner product Length of a vector Distance between vectors Orthogonal vectors Section 6 2 Orthogonal Sets Orthogonal sets Ex Show the set cid 3422 cid 3430 cid 3434 cid 3430 cid 3434 cid 3430 cid 3434 cid 3426 is an orthogonal set 1 0 5 2 0 1 2 5 5 2 1 0 Orthonormal sets Math 220 Fall 2023 Lecture Notes Page 8 Orthogonal projection onto a line spanned by cid 4652 cid 3548 cid 3048 cid 4652 cid 4652 The error vector is 1 cid 3548 is the point on the line closest to 2 is orthogonal to the line 3 The distance from to the line is dist Section 6 3 Orthogonal Projections Orthogonal projection onto a subspace Let cid 2869 cid 4652 cid 4652 cid 4652 cid 4652 cid 2870 cid 4652 cid 4652 cid 4652 cid 4652 cid 3043 cid 4652 cid 4652 cid 4652 cid 4652 be an orthogonal basis for a subspace Then cid 3548 cid 3024 The error vector is 1 cid 3548 is the point in closest to 2 is orthogonal to 3 The distance from to is dist Ex Let span cid 3422 cid 3430 cid 3434 cid 3430 cid 3434 cid 3426 and cid 3430 cid 3434 1 0 5 2 0 1 2 5 2 8 4 6 1 Find the closest point to in the subspace Math 220 Fall 2023 Lecture Notes Page 9 2 Find the distance from to Section 6 4 The Gram Schmidt Process The Gram Schmidt orthogonalization algorithm Given a basis cid 3419 cid 2869 cid 4652 cid 4652 cid 4652 cid 4652 cid 3043 cid 4652 cid 4652 cid 4652 cid 4652 cid 3423 for a subspace we can create an orthogonal basis cid 2869 cid 4652 cid 4652 cid 4652 cid 4652 cid 3043 cid 4652 cid 4652 cid 4652 cid 4652 for using the Gram Schmidt algorithm QR factorization is an matrix with linearly independent columns is an matrix whose columns are an orthonormal basis for Col found using Gram Schmidt and normalization on the columns of cid 3021 is an upper triangular invertible matrix with positive entries on the diagonal Ex Find a QR factorization of cid 3430 3 1 2 4 1 1 1 0 cid 3434 Math 220 Fall 2023 Lecture Notes Page 10 Section 6 5 Least Squares Problems If cid 4652 is inconsistent but a solution is still demanded we instead find the least squares solution cid 3548 which satisfies the normal equations cid 3021 cid 3021 cid 4652 Ex Let cid 3429 cid 3433 and cid 4652 cid 3429 2 1 2 0 2 …